Group Td Research Articles

Irreducible tensor operators and operations over them in the theory of molecular spectra

The explicit shape of symmetrized powers of three-dimensional representations of the groups Td and Oh is found in general form. Equations are obtained for calculating the vibrational commutators and matrix elements of irreducible tensor operators. Programs have been developed (in the C and FORTRAN languages) to remove the multiple-valued property of the vibrational-rotational Hamiltonian of spherical top molecules (CH4, SiH4, GeH4, SF6, and so on).

The group theory of trigonal and tetrahedral molecule rotations in trigonal and tetrahedral crystal fields

The permutation symmetry groups introduced by Longuet-Higgins are employed to solve the group theory for molecules of C3v and Td symmetry rotating in crystal fields of C3v and Td symmetry. The qualitative appearances of the energy level diagrams for the groups (C3v, [Cbar]3v), (C3v, [Tbar]d) and (Td, [Cbar]3v) are deduced for the range of barrier heights from the free-rotor limits to the pinned limits, for the ground state multiplet only. The correlations between the groups derived here and the group (Td, [Tbar]d) treated by Miller and Decius are considered and the alteration in the ground state splitting pattern in going from a tetrahedral molecule in a tetrahedral field through to a trigonal molecule in a trigonal field is illustrated. The group (C3v, [Cbar]3v) is treated in detail and applied to the tunnelling states of monodeuteromethane adsorbed on the surface of graphite. The rotational Hamiltonian matrix is constructed in the pocket state formalism and diagonalized using the character table for (C3v, [Cbar]3v). The quantitative appearance of the energy level diagram for the ground librational state of CH3D on graphite is shown.

Electronic structure and deep impurity levels in [111] GaAs/AlxGa1−xAs semiconductor superlattices

A theory of the electronic structures of periodic N1×N2 GaAs/AlxGa1−xAs superlattices grown along the [111] direction is presented. Deep levels associated with s- and p-bonded substitutional impurities in these superlattices are also predicted. It is found that: (i) in contrast with [001] superlattices, [111] superlattices are almost always direct band-gap superlattices. (ii) The [111] superlattices exhibit weaker quantum-well confinement than the corresponding [001] superlattices. (iii) As the thickness, t(GaAs), of each GaAs layer is reduced below a critical value (tc ≂13 Å or N1≂4 for x=0.7iii), common shallow donor impurities such as Si cease donating electrons to the conduction band and instead become deep traps. For [111] superlattices tc is smaller than the corresponding tc for [001] superlattices. The fundamental band gap and the band edges of the superlattice, and hence the ionization energies of deep levels, depend strongly on the layer thickness t(GaAs) but only weakly on t(AlxGa1−xAs). The T2- and A1-derived deep levels (of the bulk point group Td) are split and shifted, respectively, near a GaAs/AlxGa1−xAs interface: the p-like T2 level splits into an a1 ( pσ-like) level and a doubly degenerate e ( pπ -like) level of the point group for any general superlattice site (C3v), whereas the s-like A1 bulk level becomes an a1 (s-like) level of C3v. The order of magnitude of the shifts and splittings of deep levels at a GaAs/AlxGa1−xAs interface is less than 0.1 eV, depends on x, and becomes very small for impurities more than ≂3 atomic planes away from an interface. These predictions are based on a periodic superslab calculation for unit superslabs with total thickness t(GaAs)+t(AlxGa1−xAs) as large as 65.3 Å or N1+N2=20 two-atom-thick layers. The Hamiltonian is a tight-binding model in a hybrid basis that is a generalization of the Vogl model and properly accounts for the nature of interfacial bonds. The deep levels are computed using the Hjalmarson et al. theory [Phys. Rev. Lett. 44, 810 (1980)] and the special points method.

Construction of linearly independent relativistic symmetry orbitals for finite double‐point groups including time reversal symmetry

AbstractA formalism is developed for the construction of relativistic symmetry‐adapted molecular basis functions under consideration of time reversal invariance. The theory is applicable to the finite double point groups Cn, Cnh, Sn, Cnv, Dn, Dnd, Dnh, T, Th, Td, O, and Oh. It is based on the LCAO method. A projection operator technique is employed to construct molecular symmetry orbitals from atomic orbitals. The search for linearly independent basis function is simplified by means of group theoretical considerations.

Photoelectron spectroscopic studies of polyatomic molecules. II. Ionisation in the a1orbital of a Tdmolecule

For pt.I see ibid., vol.20, p.3405-15 (1987). A recently developed theory is used to study photoionisation in bound orbitals belonging to the a1 irreducible representation of those polyatomic molecules which transform like the point symmetry group Td. The angle-resolved photoelectron spectra (ARPES), currently being measured, of CH4, CF4, CCl4, SiCl4, and GeCl4 have been discussed. Unlike in the case of diatomics, even the rotationally averaged angular distributions of electrons ejected in photoionisation of these non-linear targets are found to provide a very sensitive test of theoretical models.

A new method of constructing the symmetry-adapted linear combinations based on the correspondence theorem and induced representations

The theory of induced representations is incorporated into the method of constructing symmetry-adapted linear combinations for a symmetry group based on the correspondence theorem developed previously. The theory is effective for a highly symmetric molecule with a set of equivalent radicals, each of which again consists of a number of equivalent atoms. An illustrative example is given to construct the internal symmetry coordinates of molecular vibrations of C(CH3)4 with the frozen configuration belonging to the point group Td.

Invariant Harmonic Oscillator Functions. I. The Cubic Groups Td, O, and Oh

AbstractExcited energy states of quantum mechanical two‐ or three‐dimensional harmonic oscillators are of interest in Jahn‐Teller calculations and rotational tunneling systems. A general method is given to calculate symmetry adapted oscillator functions by numerical computation. The basic relations are formulated in the occupation number representation. Therefore advantage of group properties like isomorphism and direct product groups can be taken easily. Results are given for two‐ and three‐dimensional irreducible representations of the cubic groups Oh, O, and Td.

Orbit space and minima of Landau–Higgs potentials

We establish a connection between recent propositions for minimization of Higgs potentials in an orbit space and related treatments in the Landau theory of phase transitions. Starting from isotropy groups we give a simple and systematic procedure for determining a stratification in an orbit space. As an illustration we treat the point groups Oh and Td which occur in the Higgs mechanism for SO7 and SU4 adjoint representations.

Point-group selection rules for ΔS=2 multiplicity change: Application to catalysis and organo-transition metal reaction

Electron spin–spin and second-order spin-orbit interaction operators are expanded as products of irreducible representations of symmetry point groups (Oh, Td, D5d, D6d, and C4v). From the transformation of the separated orbit and of the spin part, the selection rules for off-diagonal matrix elements may be deduced by taking direct products of the ‘‘initial’’ and ‘‘final’’ states. The special ΔMl selection rule for the orbital part of spin–spin interaction after expansion is also discussed. Emphasis is given to the ΔS=2 change connected by these operators. Possible examples of ΔS=2 change in d4, d5, and d6 configurations under the above mentioned point groups are given. As illustrations of the selection rules, the matrix elements for ΔS=2 and ΔMs=2 for these configurations are evaluated in the decoupled representation and given in terms of common parameters. The relevance of these multiplicity change to catalysis and reaction of organo-transition metal complexes is briefly alluded to.

Group‐theoretical study of light‐induced electron‐spin polarization at the Γ‐point of cubic crystals

AbstractSelection rules are calculated for the excitation of light‐induced spin‐polarized electrons at the center of the Brillouin zone of cubic crystals. The derivation of such rules is based on an application of the generalized Wigner‐Eckart theorem and tabulated coupling coefficients for unitary point groups. For almost all transitions a net electron‐spin polarization less than 50% is achieved. This is mainly due to the existence of bases which are linear combinations of spin‐up and spin‐down states. Tables are presented for the groups Oh, O, and Td. The connection of the results for Γ to those of the high‐symmetry lines is discussed.

Vibrationally induced nuclear quadrupole coupling in tetrahedral and octahedral molecules

Hyperfine splittings arising from the presence of a quadrupolar nucleus at the center of a molecule belonging to the point group Td or Oh (e.g., 189OsO4 or 235UF6) are symmetry forbidden to a high degree of approximation. Nevertheless, quadrupole splittings can be induced by either vibrational or rotational distortions of the molecule, i.e., by distortions similar to those responsible for inducing electric dipole moments in Td molecules. Such hyperfine splittings have recently been observed in several laboratories using laser saturation spectroscopy. In this paper we investigate theoretically the quadrupole splittings induced by excitation of doubly and triply degenerate vibrations in Td and Oh molecules. We find that much of the vibration–rotation formalism already present in the methane literature can be applied with only minor changes to the induced-quadrupole-coupling problem, and that rather simple theoretical relationships can be derived between the quadrupole splitting and the tetrahedral or octahedral vibration–rotation splitting of a given level. Values for the scalar and tensor contribution to the quadrupole coupling constant have been derived from the experimental data for 189OsO4 reported by Bordé et al.

Développement complet de l'hamiltonien de vibration–rotation adapté à l'étude des interactions dans les molécules toupies sphériques. Application aux bandes ν2 et ν4 de 12CH4

Using an unsymmetrized coupling scheme in the group Td, we determine all the vibration–rotation operators of the Hamiltonian of XY4 molecules, including all possible interactions, up to any order of approximation. We define a basis of Hamiltonian matrices of which the matrix elements are functions of coupling coefficients of the group chain [Formula: see text] only. Thus, we develop a general formalism available for any vibrational sublevels of whatever symmetry. The parameters relative to the different vibrational sublevels are known linear combinations of the coefficients of the Hamiltonian expansion. From these, we deduce simple relations between the parameters associated with the ground state, the fundamentals, and the harmonic and combination bands.We apply this formalism to the study of the Coriolis interaction between ν2 and ν4 of CH4. With only 21 parameters for the two bands, we obtain a standard deviation of 34 mK for 251 Raman transitions of ν2 and 20 mK for 243 ir transitions of ν4.

Localized states in semiconductors: isocoric impurities in Si and Ge

A formalism is developed within the multi-valley effective mass approximation for the calculation of the groundstate energies for isocoric donor impurities in silicon and germanium taking into account the ellipsoidal shape of their energy surfaces. For the impurity potential a general screened Coulomb form is considered and its parameters are determined from the k dependent dielectric functions of Si and Ge. Variational trial wavefunctions, belonging to an irreducible representation of the tetrahedral point group Td of the Hamiltonian, are used. The groundstate energies as expressed in terms of the effective masses, the potential parameters and the location k0 of the conduction band minima, are then determined by minimization. The results for Si are in good agreement with experiments; however for Ge the predicted E(A1)-E(T2) splitting is rather too small, -0.5 to -0.7 meV compared with the observed value of -4.23 meV.

Molecular Rovibronic Symmetries in Electric and Magnetic Fields

The structures of the symmetry groups for the rovibronic levels of a molecule in a homogeneous electric or magnetic field are derived, and the symmetry classification of the levels in terms of the representations and corepresentations of these groups is discussed. Specific results are given for molecules of the point groups C3, C2v, C3v, D2d, and Td in an electric field. Symmetry in combined electric and magnetic fields and the inclusion of nuclear spins are considered briefly.

Normal coordinates for a non-planar molecule XY4in a Cartesian basis

A unique set of normal coordinates is constructed in a Cartesian basis for a non-planar molecule XY4 belonging to the group Td, using the method proposed by Sheth (1975), and hence establishing the usefulness of the method for cubic groups.

Vibrational spectra of zirconium tetrahydroborate and related molecules

The i.r. spectra of Zr(BH4)4, Zr(BD4)4, Hf(BH4)4, Hf(BD4)4, and U(BH4)4 have been interpreted in conjunction with the Raman spectra of benzene solutions of Zr(BH4)4 and Hf(BH4)4. In all cases the results are strongly indicative of a triple hydrogen bridge structure of symmetry point group Td.

Uniaxial-strain effects on n = 1 free-exciton and free-carrier lines in GaAs

Photoluminescence at liquid-helium temperature from free excitons in GaAs has been studied for zero strain and for [100] stresses from 33 to 500 kg cm-2. At zero strain (group Td) there are three exchange-split exciton lines Λ5 (J = 1, allowed), and Λ3 and Λ4 (J = 2, forbidden, and at lower energy than J = 1). The Λ5 line is strain split by [100] stress χ (group D2d) into a higher-energy line X2 of σ polarization (⊥ stress) and a lower- energy line X1a of π polarization (∥ stress). The Λ4 lines gives, with this strain, an allowed, lower-energy line X1b of σ polarization. The exchange splitting E between Λ5 and Λ4 is 0.37 ± 0.04 meV. Theory predicts relative strengths X1a:X1b:X2 of π:0:σ for E ≪ strain splitting Δ hv, changing to π:σ/2:σ/2 for Δ hv ≪ E, and our data are consistent. The deformation potential coefficients for the Λ5 exciton states are a = -9.76 ± 0.03 eV and b = -1.52 ± 0.06 eV. The dependence of EG on χ has been determined and found to be more rapid than that for X1a. The value for G(1) thus derived is 4.4 ± 0.04 meV and for EG we find 1.51996 eV ± 0.04 meV for one sample.

Electroabsorption by Substitutional Copper Impurities in GaAs

Previously, Hall effect and other thermal activation measurements have suggested that a copper atom substituted on a gallium site in GaAs is a double acceptor with its first ionization level at 0.145 eV and its second between 0.44 and 0.49 eV above the valence band. Also, previous photoluminescence measurements have revealed broad lines attributable to copper at the Stokes-shifted positions 0.155 eV and between 0.49 and 0.53 eV. Using an electroabsorption technique, we have studied the spectrum of substitutional copper in GaAs. We find an absorption line associated with a level at 0.463 eV above the valence band for temperatures of 103° and 300°K. We conclude from the selection rules that the wavefunction associated with this level transforms as the Γ6 representation of the double group Td. Except for a weaker additional absorption line (which we do not yet understand) occurring at 0.454 eV in a single p-type sample, we were able to find no other lines. These results supplement earlier work on the 0.145 eV state. We propose a simple model which we feel describes the substitutional copper impurity and apply it to our measurements on the 0.463 eV state and our earlier work on the 0.145 eV state. While this model is not adequate for a first-principles calculation, it does provide a framework in which to understand and discuss our results.

Infrared Spectra of Cubic and Tetragonal Manganese Ferrites

AbstractThe infrared absorption spectra in the range from 200 to 700 cm−1 of cubic and tetragonal manganese ferrites, MnxFe3–xO4, are reported. An analysis of the spectra is made with a model by which the lattice vibrations are classified into species of the point group D2 d on the analogy of the results for gas molecules of equivalent symmetry. For cubic spinels with manganese concentrations x ≦ 1.0 the classification can be made according to the model of Waldron into the species of the point group Td. However, at 335 cm−1 an absorption has been observed which is out of the scope of the Td or D2 d classification, and which is caused by the substitution of a part of the octahedral Fe3+ ions by Mn2+ ions. The IR spectra of manganese ferrites with compositions 1 < x ≦ 3 can be analysed with the D2 d classification. The spectra of these materials do not change suddenly with the manganese concentration or with the crystal structure. From the splitting of the absorption bands in cubic ferrites with x > 1 it is concluded that there are local tetragonal distortions originating from the Jahn‐Teller effect of the octahedral Mn3+ ions.

Infrared Absorption Spectrum of Silicon Dioxide

The absorption spectra of fused silica and α-quartz were investigated in the 2600 to 50 cm−1 wave number region. Only part of the absorption frequency bands of the crystal quartz appeared in the fused state and some of them were shifted to higher wave numbers. The principal absorption frequencies in the fused silica were at 1126, 809, 452, and 200 cm−1 It is proposed that the infrared spectrum of the fused silica results from two vibrating units, of point group Td and D3h, simultaneously. In the vitreous state, some of the Si-O-Si bonds are disrupted allowing random orientation of the tetrahedral SiO4 groups (point group Td). As a consequence of the formation of nonbridging oxygen, the force constant will be increased, as indicated by the shift of the vibrational frequencies to higher values. For every nonbridging oxygen atom formed, the silicon atom previously connected to it would be left with a positive hole. The point group D3h is due to such a silicon atom and its v3 and v4 modes coincide with the v3 and v4 modes of the Td point group.